I found this problem in one of those foreign Mathematical Olympiad books. The solution is provided, but even then I still can't grasp the steps that lead to it.
We have a triangle ABC as shown in the figure, with point E selected on side AB in such a way that AE : EB = 1:3, in other words, the ratio of the length of segment AE to the length of segment EB is 1 to 3. Draw the line segment CE.
We then have a point D selected on side BC so that CD : DB = 1:2. Then draw a line segment from A to D. This gives a point F, which is the intersection of segments AD and CE.
Solve for (EF/FC) + (AF/FD).